#7645828

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4279 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(2,5)(6,8)(10,16)(14,18)(17,20)(19,24)(26,30)(27,33)(32,38)(34,35);(1,2)(3,6)(4,5)(7,8)(9,10,15,16)(11,20,22,17)(12,18,21,14)(13,19,23,24);(9,11,15,22)(10,17,16,20)(12,13,21,23)(14,19,18,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(9,12)(10,18)(11,13)(14,16)(15,21)(17,24)(19,20)(22,23)(25,27,29,33)(26,31,30,37)(28,35,36,34)(32,40,38,39);(9,13)(10,19)(11,21)(12,22)(14,20)(15,23)(16,24)(17,18)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40);(1,3,4,7)(2,6,5,8)(9,14,15,18)(10,12,16,21)(11,19,22,24)(13,20,23,17)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553024);